Find the volume of the solid

Question

Using disks or washers, find the volume of the solid obtained by rotating the region bounded by the curves y=√(x-1), y=0, x=2, and x=5 about the x-axis 

Answer 1

When this curve rotates about x axis it will look like a trapeziod with radius of 1 unit (√(2-1))at one end and radius of 4 units (√(5-1)) at another end  with height of 3 units (5-2=3_like a chopped off cone.

 

You can use calculus

 

volume of an infinite thin disc of thixkness dx with radius r = πr²

r varies as y=√(x-1)

so volume of infinite thin disc = π(√(x-1))²dx = π (x-1)dx

 

Total volume = integrate this expression from x = 2 to x=5

 

total vulome = Π( x²/2  -  x) with x varying from 2 to 5

 

so total volume = Π ((5²/2  - 5) - (2²/2 - 2)) = 7.5Π = 23.55 cubic units

:)